2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.6%

Time: 17.7s

Alternatives: 11

Speedup: 205.0×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(t\_0 + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(\left(t\_0 \cdot -0.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right) - t\_0\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + 1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    eps
    (+
     t_0
     (+
      (*
       eps
       (+
        (*
         eps
         (-
          0.3333333333333333
          (-
           (-
            (* t_0 -0.3333333333333333)
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))
           t_0)))
        (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
      1.0)))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * (t_0 + ((eps * ((eps * (0.3333333333333333 - (((t_0 * -0.3333333333333333) - (pow(sin(x), 4.0) / pow(cos(x), 4.0))) - t_0))) + ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))))) + 1.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    code = eps * (t_0 + ((eps * ((eps * (0.3333333333333333d0 - (((t_0 * (-0.3333333333333333d0)) - ((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0))) - t_0))) + ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))) + 1.0d0))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0);
	return eps * (t_0 + ((eps * ((eps * (0.3333333333333333 - (((t_0 * -0.3333333333333333) - (Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0))) - t_0))) + ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))))) + 1.0));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	return eps * (t_0 + ((eps * ((eps * (0.3333333333333333 - (((t_0 * -0.3333333333333333) - (math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0))) - t_0))) + ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))) + 1.0))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(t_0 + Float64(Float64(eps * Float64(Float64(eps * Float64(0.3333333333333333 - Float64(Float64(Float64(t_0 * -0.3333333333333333) - Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))) - t_0))) + Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))))) + 1.0)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * (t_0 + ((eps * ((eps * (0.3333333333333333 - (((t_0 * -0.3333333333333333) - ((sin(x) ^ 4.0) / (cos(x) ^ 4.0))) - t_0))) + ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))))) + 1.0));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(t$95$0 + N[(N[(eps * N[(N[(eps * N[(0.3333333333333333 - N[(N[(N[(t$95$0 * -0.3333333333333333), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. tan-sum 61.8%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. div-inv 61.8%

      \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   3. fmm-def 61.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

4. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
5. Step-by-step derivation

   1. fmm-undef 61.8%

      \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]

6. Simplified 61.8%

   \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]

7. Taylor expanded in eps around 0 99.2%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

8. Final simplification 99.2%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -0.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + 1\right)\right) \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := {\sin x}^{2}\\ \varepsilon \cdot \left(\left(\varepsilon \cdot \left(\frac{\sin x \cdot \left(\frac{t\_1}{{\cos x}^{2}} + 1\right)}{\cos x} - \varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, t\_0 + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, t\_0, -0.5 + t\_0 \cdot -0.5\right)\right)\right)\right) + 1\right) + \frac{t\_1}{\frac{\cos \left(x \cdot 2\right) + 1}{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (pow (sin x) 2.0)))
   (*
    eps
    (+
     (+
      (*
       eps
       (-
        (/ (* (sin x) (+ (/ t_1 (pow (cos x) 2.0)) 1.0)) (cos x))
        (*
         eps
         (+
          0.16666666666666666
          (fma
           -1.0
           (+ t_0 (pow (tan x) 4.0))
           (fma 0.16666666666666666 t_0 (+ -0.5 (* t_0 -0.5))))))))
      1.0)
     (/ t_1 (/ (+ (cos (* x 2.0)) 1.0) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = pow(sin(x), 2.0);
	return eps * (((eps * (((sin(x) * ((t_1 / pow(cos(x), 2.0)) + 1.0)) / cos(x)) - (eps * (0.16666666666666666 + fma(-1.0, (t_0 + pow(tan(x), 4.0)), fma(0.16666666666666666, t_0, (-0.5 + (t_0 * -0.5)))))))) + 1.0) + (t_1 / ((cos((x * 2.0)) + 1.0) / 2.0)));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = sin(x) ^ 2.0
	return Float64(eps * Float64(Float64(Float64(eps * Float64(Float64(Float64(sin(x) * Float64(Float64(t_1 / (cos(x) ^ 2.0)) + 1.0)) / cos(x)) - Float64(eps * Float64(0.16666666666666666 + fma(-1.0, Float64(t_0 + (tan(x) ^ 4.0)), fma(0.16666666666666666, t_0, Float64(-0.5 + Float64(t_0 * -0.5)))))))) + 1.0) + Float64(t_1 / Float64(Float64(cos(Float64(x * 2.0)) + 1.0) / 2.0))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(N[(N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(t$95$1 / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(0.16666666666666666 + N[(-1.0 * N[(t$95$0 + N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * t$95$0 + N[(-0.5 + N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(t$95$1 / N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 61.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.2%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. unpow2 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\cos x \cdot \cos x}}\right) \]

   2. cos-mult 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right) \]

5. Applied egg-rr 99.2%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right) \]
6. Step-by-step derivation

   1. +-commutative 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\color{blue}{\cos \left(x - x\right) + \cos \left(x + x\right)}}{2}}\right) \]

   2. +-inverses 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\cos \color{blue}{0} + \cos \left(x + x\right)}{2}}\right) \]

   3. cos-0 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\color{blue}{1} + \cos \left(x + x\right)}{2}}\right) \]

   4. count-2 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(2 \cdot x\right)}}{2}}\right) \]

   5. *-commutative 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(x \cdot 2\right)}}{2}}\right) \]

7. Simplified 99.2%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}}\right) \]
8. Step-by-step derivation

   1. fma-define 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \color{blue}{\mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

9. Applied egg-rr 99.2%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \color{blue}{\mathsf{fma}\left(-1, \frac{1 + {\tan x}^{2}}{1} \cdot {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)}\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]
10. Step-by-step derivation

    1. /-rgt-identity 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \color{blue}{\left(1 + {\tan x}^{2}\right)} \cdot {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    2. +-commutative 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \color{blue}{\left({\tan x}^{2} + 1\right)} \cdot {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    3. *-commutative 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \color{blue}{{\tan x}^{2} \cdot \left({\tan x}^{2} + 1\right)}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    4. +-commutative 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} \cdot \color{blue}{\left(1 + {\tan x}^{2}\right)}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    5. distribute-lft-in 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \color{blue}{{\tan x}^{2} \cdot 1 + {\tan x}^{2} \cdot {\tan x}^{2}}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    6. *-rgt-identity 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \color{blue}{{\tan x}^{2}} + {\tan x}^{2} \cdot {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    7. pow-sqr 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + \color{blue}{{\tan x}^{\left(2 \cdot 2\right)}}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    8. metadata-eval 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{\color{blue}{4}}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    9. fma-undefine 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \color{blue}{{\tan x}^{2} \cdot 0.16666666666666666 + \left(1 + {\tan x}^{2}\right) \cdot -0.5}\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    10. *-commutative 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \color{blue}{0.16666666666666666 \cdot {\tan x}^{2}} + \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    11. fma-define 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \color{blue}{\mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)}\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    12. +-commutative 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \color{blue}{\left({\tan x}^{2} + 1\right)} \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    13. *-commutative 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \color{blue}{-0.5 \cdot \left({\tan x}^{2} + 1\right)}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    14. +-commutative 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, -0.5 \cdot \color{blue}{\left(1 + {\tan x}^{2}\right)}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    15. distribute-lft-in 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \color{blue}{-0.5 \cdot 1 + -0.5 \cdot {\tan x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

11. Simplified 99.2%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \color{blue}{\mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, -0.5 + -0.5 \cdot {\tan x}^{2}\right)\right)}\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

12. Final simplification 99.2%

    \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \left(\frac{\sin x \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}{\cos x} - \varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, -0.5 + {\tan x}^{2} \cdot -0.5\right)\right)\right)\right) + 1\right) + \frac{{\sin x}^{2}}{\frac{\cos \left(x \cdot 2\right) + 1}{2}}\right) \]
13. Add Preprocessing

Alternative 3: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ \varepsilon \cdot \left(\frac{t\_0}{\frac{\cos \left(x \cdot 2\right) + 1}{2}} + \left(\varepsilon \cdot \left(\frac{\sin x \cdot \left(\frac{t\_0}{{\cos x}^{2}} + 1\right)}{\cos x} + \varepsilon \cdot 0.3333333333333333\right) + 1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0)))
   (*
    eps
    (+
     (/ t_0 (/ (+ (cos (* x 2.0)) 1.0) 2.0))
     (+
      (*
       eps
       (+
        (/ (* (sin x) (+ (/ t_0 (pow (cos x) 2.0)) 1.0)) (cos x))
        (* eps 0.3333333333333333)))
      1.0)))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	return eps * ((t_0 / ((cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * (((sin(x) * ((t_0 / pow(cos(x), 2.0)) + 1.0)) / cos(x)) + (eps * 0.3333333333333333))) + 1.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin(x) ** 2.0d0
    code = eps * ((t_0 / ((cos((x * 2.0d0)) + 1.0d0) / 2.0d0)) + ((eps * (((sin(x) * ((t_0 / (cos(x) ** 2.0d0)) + 1.0d0)) / cos(x)) + (eps * 0.3333333333333333d0))) + 1.0d0))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	return eps * ((t_0 / ((Math.cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * (((Math.sin(x) * ((t_0 / Math.pow(Math.cos(x), 2.0)) + 1.0)) / Math.cos(x)) + (eps * 0.3333333333333333))) + 1.0));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	return eps * ((t_0 / ((math.cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * (((math.sin(x) * ((t_0 / math.pow(math.cos(x), 2.0)) + 1.0)) / math.cos(x)) + (eps * 0.3333333333333333))) + 1.0))

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	return Float64(eps * Float64(Float64(t_0 / Float64(Float64(cos(Float64(x * 2.0)) + 1.0) / 2.0)) + Float64(Float64(eps * Float64(Float64(Float64(sin(x) * Float64(Float64(t_0 / (cos(x) ^ 2.0)) + 1.0)) / cos(x)) + Float64(eps * 0.3333333333333333))) + 1.0)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	tmp = eps * ((t_0 / ((cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * (((sin(x) * ((t_0 / (cos(x) ^ 2.0)) + 1.0)) / cos(x)) + (eps * 0.3333333333333333))) + 1.0));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(N[(t$95$0 / N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(t$95$0 / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.2%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. unpow2 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\cos x \cdot \cos x}}\right) \]

   2. cos-mult 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right) \]

5. Applied egg-rr 99.2%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right) \]
6. Step-by-step derivation

   1. +-commutative 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\color{blue}{\cos \left(x - x\right) + \cos \left(x + x\right)}}{2}}\right) \]

   2. +-inverses 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\cos \color{blue}{0} + \cos \left(x + x\right)}{2}}\right) \]

   3. cos-0 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\color{blue}{1} + \cos \left(x + x\right)}{2}}\right) \]

   4. count-2 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(2 \cdot x\right)}}{2}}\right) \]

   5. *-commutative 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(x \cdot 2\right)}}{2}}\right) \]

7. Simplified 99.2%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}}\right) \]

8. Taylor expanded in x around 0 99.1%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\color{blue}{0.3333333333333333 \cdot \varepsilon} - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

9. Final simplification 99.1%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{\frac{\cos \left(x \cdot 2\right) + 1}{2}} + \left(\varepsilon \cdot \left(\frac{\sin x \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}{\cos x} + \varepsilon \cdot 0.3333333333333333\right) + 1\right)\right) \]
10. Add Preprocessing

Alternative 4: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ \varepsilon \cdot \left(\frac{t\_0}{\frac{\cos \left(x \cdot 2\right) + 1}{2}} + \left(1 - \varepsilon \cdot \left(\sin x \cdot \frac{-1 - \frac{t\_0}{{\cos x}^{2}}}{\cos x}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0)))
   (*
    eps
    (+
     (/ t_0 (/ (+ (cos (* x 2.0)) 1.0) 2.0))
     (-
      1.0
      (* eps (* (sin x) (/ (- -1.0 (/ t_0 (pow (cos x) 2.0))) (cos x)))))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	return eps * ((t_0 / ((cos((x * 2.0)) + 1.0) / 2.0)) + (1.0 - (eps * (sin(x) * ((-1.0 - (t_0 / pow(cos(x), 2.0))) / cos(x))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin(x) ** 2.0d0
    code = eps * ((t_0 / ((cos((x * 2.0d0)) + 1.0d0) / 2.0d0)) + (1.0d0 - (eps * (sin(x) * (((-1.0d0) - (t_0 / (cos(x) ** 2.0d0))) / cos(x))))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	return eps * ((t_0 / ((Math.cos((x * 2.0)) + 1.0) / 2.0)) + (1.0 - (eps * (Math.sin(x) * ((-1.0 - (t_0 / Math.pow(Math.cos(x), 2.0))) / Math.cos(x))))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	return eps * ((t_0 / ((math.cos((x * 2.0)) + 1.0) / 2.0)) + (1.0 - (eps * (math.sin(x) * ((-1.0 - (t_0 / math.pow(math.cos(x), 2.0))) / math.cos(x))))))

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	return Float64(eps * Float64(Float64(t_0 / Float64(Float64(cos(Float64(x * 2.0)) + 1.0) / 2.0)) + Float64(1.0 - Float64(eps * Float64(sin(x) * Float64(Float64(-1.0 - Float64(t_0 / (cos(x) ^ 2.0))) / cos(x)))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	tmp = eps * ((t_0 / ((cos((x * 2.0)) + 1.0) / 2.0)) + (1.0 - (eps * (sin(x) * ((-1.0 - (t_0 / (cos(x) ^ 2.0))) / cos(x))))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(N[(t$95$0 / N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(eps * N[(N[Sin[x], $MachinePrecision] * N[(N[(-1.0 - N[(t$95$0 / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.2%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. unpow2 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\cos x \cdot \cos x}}\right) \]

   2. cos-mult 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right) \]

5. Applied egg-rr 99.2%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right) \]
6. Step-by-step derivation

   1. +-commutative 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\color{blue}{\cos \left(x - x\right) + \cos \left(x + x\right)}}{2}}\right) \]

   2. +-inverses 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\cos \color{blue}{0} + \cos \left(x + x\right)}{2}}\right) \]

   3. cos-0 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\color{blue}{1} + \cos \left(x + x\right)}{2}}\right) \]

   4. count-2 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(2 \cdot x\right)}}{2}}\right) \]

   5. *-commutative 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(x \cdot 2\right)}}{2}}\right) \]

7. Simplified 99.2%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}}\right) \]

8. Taylor expanded in eps around 0 99.1%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}}\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]
9. Step-by-step derivation

   1. associate-/l* 99.1%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}}\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

   2. associate-/l* 99.1%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \color{blue}{\left(\sin x \cdot \frac{1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}{\cos x}\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

   3. sub-neg 99.1%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\sin x \cdot \frac{\color{blue}{1 + \left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

   4. mul-1-neg 99.1%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\sin x \cdot \frac{1 + \left(-\color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

   5. remove-double-neg 99.1%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\sin x \cdot \frac{1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

10. Simplified 99.1%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \left(\sin x \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{\cos x}\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

11. Final simplification 99.1%

    \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{\frac{\cos \left(x \cdot 2\right) + 1}{2}} + \left(1 - \varepsilon \cdot \left(\sin x \cdot \frac{-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}}{\cos x}\right)\right)\right) \]
12. Add Preprocessing

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot x\right) + 1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
   (+ (+ (* 0.3333333333333333 (pow eps 2.0)) (* eps x)) 1.0))))

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + (((0.3333333333333333 * pow(eps, 2.0)) + (eps * x)) + 1.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + (((0.3333333333333333d0 * (eps ** 2.0d0)) + (eps * x)) + 1.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + (((0.3333333333333333 * Math.pow(eps, 2.0)) + (eps * x)) + 1.0));
}

Python

def code(x, eps):
	return eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + (((0.3333333333333333 * math.pow(eps, 2.0)) + (eps * x)) + 1.0))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + Float64(Float64(Float64(0.3333333333333333 * (eps ^ 2.0)) + Float64(eps * x)) + 1.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + (((0.3333333333333333 * (eps ^ 2.0)) + (eps * x)) + 1.0));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(eps * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.2%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

4. Taylor expanded in x around 0 98.5%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot x\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

5. Final simplification 98.5%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot x\right) + 1\right)\right) \]
6. Add Preprocessing

Alternative 6: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{\frac{\cos \left(x \cdot 2\right) + 1}{2}} + \left(\varepsilon \cdot \left(x + \varepsilon \cdot 0.3333333333333333\right) + 1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (/ (pow (sin x) 2.0) (/ (+ (cos (* x 2.0)) 1.0) 2.0))
   (+ (* eps (+ x (* eps 0.3333333333333333))) 1.0))))

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / ((cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * (x + (eps * 0.3333333333333333))) + 1.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((sin(x) ** 2.0d0) / ((cos((x * 2.0d0)) + 1.0d0) / 2.0d0)) + ((eps * (x + (eps * 0.3333333333333333d0))) + 1.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps * ((Math.pow(Math.sin(x), 2.0) / ((Math.cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * (x + (eps * 0.3333333333333333))) + 1.0));
}

Python

def code(x, eps):
	return eps * ((math.pow(math.sin(x), 2.0) / ((math.cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * (x + (eps * 0.3333333333333333))) + 1.0))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64((sin(x) ^ 2.0) / Float64(Float64(cos(Float64(x * 2.0)) + 1.0) / 2.0)) + Float64(Float64(eps * Float64(x + Float64(eps * 0.3333333333333333))) + 1.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((sin(x) ^ 2.0) / ((cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * (x + (eps * 0.3333333333333333))) + 1.0));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[(x + N[(eps * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.2%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. unpow2 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\cos x \cdot \cos x}}\right) \]

   2. cos-mult 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right) \]

5. Applied egg-rr 99.2%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right) \]
6. Step-by-step derivation

   1. +-commutative 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\color{blue}{\cos \left(x - x\right) + \cos \left(x + x\right)}}{2}}\right) \]

   2. +-inverses 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\cos \color{blue}{0} + \cos \left(x + x\right)}{2}}\right) \]

   3. cos-0 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\color{blue}{1} + \cos \left(x + x\right)}{2}}\right) \]

   4. count-2 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(2 \cdot x\right)}}{2}}\right) \]

   5. *-commutative 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(x \cdot 2\right)}}{2}}\right) \]

7. Simplified 99.2%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}}\right) \]
8. Step-by-step derivation

   1. fma-define 99.2%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \color{blue}{\mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

9. Applied egg-rr 99.2%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \color{blue}{\mathsf{fma}\left(-1, \frac{1 + {\tan x}^{2}}{1} \cdot {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)}\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]
10. Step-by-step derivation

    1. /-rgt-identity 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \color{blue}{\left(1 + {\tan x}^{2}\right)} \cdot {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    2. +-commutative 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \color{blue}{\left({\tan x}^{2} + 1\right)} \cdot {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    3. *-commutative 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \color{blue}{{\tan x}^{2} \cdot \left({\tan x}^{2} + 1\right)}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    4. +-commutative 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} \cdot \color{blue}{\left(1 + {\tan x}^{2}\right)}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    5. distribute-lft-in 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \color{blue}{{\tan x}^{2} \cdot 1 + {\tan x}^{2} \cdot {\tan x}^{2}}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    6. *-rgt-identity 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \color{blue}{{\tan x}^{2}} + {\tan x}^{2} \cdot {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    7. pow-sqr 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + \color{blue}{{\tan x}^{\left(2 \cdot 2\right)}}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    8. metadata-eval 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{\color{blue}{4}}, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    9. fma-undefine 99.2%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \color{blue}{{\tan x}^{2} \cdot 0.16666666666666666 + \left(1 + {\tan x}^{2}\right) \cdot -0.5}\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    10. *-commutative 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \color{blue}{0.16666666666666666 \cdot {\tan x}^{2}} + \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    11. fma-define 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \color{blue}{\mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \left(1 + {\tan x}^{2}\right) \cdot -0.5\right)}\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    12. +-commutative 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \color{blue}{\left({\tan x}^{2} + 1\right)} \cdot -0.5\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    13. *-commutative 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \color{blue}{-0.5 \cdot \left({\tan x}^{2} + 1\right)}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    14. +-commutative 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, -0.5 \cdot \color{blue}{\left(1 + {\tan x}^{2}\right)}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

    15. distribute-lft-in 99.2%

        \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \color{blue}{-0.5 \cdot 1 + -0.5 \cdot {\tan x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

11. Simplified 99.2%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \color{blue}{\mathsf{fma}\left(-1, {\tan x}^{2} + {\tan x}^{4}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, -0.5 + -0.5 \cdot {\tan x}^{2}\right)\right)}\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

12. Taylor expanded in x around 0 98.5%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \color{blue}{\left(x + 0.3333333333333333 \cdot \varepsilon\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]

13. Final simplification 98.5%

    \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{\frac{\cos \left(x \cdot 2\right) + 1}{2}} + \left(\varepsilon \cdot \left(x + \varepsilon \cdot 0.3333333333333333\right) + 1\right)\right) \]
14. Add Preprocessing

Alternative 7: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {\tan x}^{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* eps (pow (tan x) 2.0))))

C

double code(double x, double eps) {
	return eps + (eps * pow(tan(x), 2.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * (tan(x) ** 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * Math.pow(Math.tan(x), 2.0));
}

Python

def code(x, eps):
	return eps + (eps * math.pow(math.tan(x), 2.0))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * (tan(x) ^ 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (tan(x) ^ 2.0));
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. sub-neg 98.5%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. mul-1-neg 98.5%

      \[\leadsto \varepsilon \cdot \left(1 + \left(-\color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

   3. remove-double-neg 98.5%

      \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

5. Simplified 98.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
6. Step-by-step derivation

   1. add-cbrt-cube 37.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}} \]

   2. pow1/3 35.3%

      \[\leadsto \color{blue}{{\left(\left(\left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}^{0.3333333333333333}} \]

7. Applied egg-rr 35.3%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot \varepsilon\right)}^{3}\right)}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. unpow1/3 37.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot \varepsilon\right)}^{3}}} \]

   2. rem-cbrt-cube 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot \varepsilon} \]

   3. fma-undefine 98.5%

      \[\leadsto \color{blue}{\left({\sin x}^{2} \cdot {\cos x}^{-2} + 1\right)} \cdot \varepsilon \]

   4. metadata-eval 98.5%

      \[\leadsto \left({\sin x}^{2} \cdot {\cos x}^{\color{blue}{\left(-2\right)}} + 1\right) \cdot \varepsilon \]

   5. pow-flip 98.5%

      \[\leadsto \left({\sin x}^{2} \cdot \color{blue}{\frac{1}{{\cos x}^{2}}} + 1\right) \cdot \varepsilon \]

   6. div-inv 98.5%

      \[\leadsto \left(\color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}} + 1\right) \cdot \varepsilon \]

   7. unpow2 98.5%

      \[\leadsto \left(\frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}} + 1\right) \cdot \varepsilon \]

   8. unpow2 98.5%

      \[\leadsto \left(\frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}} + 1\right) \cdot \varepsilon \]

   9. frac-times 98.5%

      \[\leadsto \left(\color{blue}{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}} + 1\right) \cdot \varepsilon \]

   10. tan-quot 98.5%

       \[\leadsto \left(\color{blue}{\tan x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \varepsilon \]

   11. tan-quot 98.5%

       \[\leadsto \left(\tan x \cdot \color{blue}{\tan x} + 1\right) \cdot \varepsilon \]

   12. pow2 98.5%

       \[\leadsto \left(\color{blue}{{\tan x}^{2}} + 1\right) \cdot \varepsilon \]

9. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\left({\tan x}^{2} + 1\right) \cdot \varepsilon} \]
10. Step-by-step derivation

    1. *-commutative 98.5%

       \[\leadsto \color{blue}{\varepsilon \cdot \left({\tan x}^{2} + 1\right)} \]

    2. distribute-rgt-in 98.5%

       \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + 1 \cdot \varepsilon} \]

    3. *-un-lft-identity 98.5%

       \[\leadsto {\tan x}^{2} \cdot \varepsilon + \color{blue}{\varepsilon} \]

11. Applied egg-rr 98.5%

    \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

12. Final simplification 98.5%

    \[\leadsto \varepsilon + \varepsilon \cdot {\tan x}^{2} \]
13. Add Preprocessing

Alternative 8: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left({\tan x}^{2} + 1\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (+ (pow (tan x) 2.0) 1.0)))

C

double code(double x, double eps) {
	return eps * (pow(tan(x), 2.0) + 1.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((tan(x) ** 2.0d0) + 1.0d0)
end function

Java

public static double code(double x, double eps) {
	return eps * (Math.pow(Math.tan(x), 2.0) + 1.0);
}

Python

def code(x, eps):
	return eps * (math.pow(math.tan(x), 2.0) + 1.0)

Julia

function code(x, eps)
	return Float64(eps * Float64((tan(x) ^ 2.0) + 1.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((tan(x) ^ 2.0) + 1.0);
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. sub-neg 98.5%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. mul-1-neg 98.5%

      \[\leadsto \varepsilon \cdot \left(1 + \left(-\color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

   3. remove-double-neg 98.5%

      \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

5. Simplified 98.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
6. Step-by-step derivation

   1. add-cbrt-cube 37.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}} \]

   2. pow1/3 35.3%

      \[\leadsto \color{blue}{{\left(\left(\left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}^{0.3333333333333333}} \]

7. Applied egg-rr 35.3%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot \varepsilon\right)}^{3}\right)}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. unpow1/3 37.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot \varepsilon\right)}^{3}}} \]

   2. rem-cbrt-cube 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot \varepsilon} \]

   3. fma-undefine 98.5%

      \[\leadsto \color{blue}{\left({\sin x}^{2} \cdot {\cos x}^{-2} + 1\right)} \cdot \varepsilon \]

   4. metadata-eval 98.5%

      \[\leadsto \left({\sin x}^{2} \cdot {\cos x}^{\color{blue}{\left(-2\right)}} + 1\right) \cdot \varepsilon \]

   5. pow-flip 98.5%

      \[\leadsto \left({\sin x}^{2} \cdot \color{blue}{\frac{1}{{\cos x}^{2}}} + 1\right) \cdot \varepsilon \]

   6. div-inv 98.5%

      \[\leadsto \left(\color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}} + 1\right) \cdot \varepsilon \]

   7. unpow2 98.5%

      \[\leadsto \left(\frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}} + 1\right) \cdot \varepsilon \]

   8. unpow2 98.5%

      \[\leadsto \left(\frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}} + 1\right) \cdot \varepsilon \]

   9. frac-times 98.5%

      \[\leadsto \left(\color{blue}{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}} + 1\right) \cdot \varepsilon \]

   10. tan-quot 98.5%

       \[\leadsto \left(\color{blue}{\tan x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \varepsilon \]

   11. tan-quot 98.5%

       \[\leadsto \left(\tan x \cdot \color{blue}{\tan x} + 1\right) \cdot \varepsilon \]

   12. pow2 98.5%

       \[\leadsto \left(\color{blue}{{\tan x}^{2}} + 1\right) \cdot \varepsilon \]

9. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\left({\tan x}^{2} + 1\right) \cdot \varepsilon} \]

10. Final simplification 98.5%

    \[\leadsto \varepsilon \cdot \left({\tan x}^{2} + 1\right) \]
11. Add Preprocessing

Alternative 9: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {x}^{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* eps (pow x 2.0))))

C

double code(double x, double eps) {
	return eps + (eps * pow(x, 2.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * (x ** 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * Math.pow(x, 2.0));
}

Python

def code(x, eps):
	return eps + (eps * math.pow(x, 2.0))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * (x ^ 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (x ^ 2.0));
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. sub-neg 98.5%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. mul-1-neg 98.5%

      \[\leadsto \varepsilon \cdot \left(1 + \left(-\color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

   3. remove-double-neg 98.5%

      \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

5. Simplified 98.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Taylor expanded in x around 0 97.5%

   \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 97.5%

      \[\leadsto \varepsilon + \color{blue}{{x}^{2} \cdot \varepsilon} \]

8. Simplified 97.5%

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \varepsilon} \]

9. Final simplification 97.5%

   \[\leadsto \varepsilon + \varepsilon \cdot {x}^{2} \]
10. Add Preprocessing

Alternative 10: 98.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(x, x, 1\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (fma x x 1.0)))

C

double code(double x, double eps) {
	return eps * fma(x, x, 1.0);
}

Julia

function code(x, eps)
	return Float64(eps * fma(x, x, 1.0))
end

Wolfram

code[x_, eps_] := N[(eps * N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. sub-neg 98.5%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. mul-1-neg 98.5%

      \[\leadsto \varepsilon \cdot \left(1 + \left(-\color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

   3. remove-double-neg 98.5%

      \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

5. Simplified 98.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
6. Step-by-step derivation

   1. add-cbrt-cube 37.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}} \]

   2. pow1/3 35.3%

      \[\leadsto \color{blue}{{\left(\left(\left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}^{0.3333333333333333}} \]

7. Applied egg-rr 35.3%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot \varepsilon\right)}^{3}\right)}^{0.3333333333333333}} \]

8. Taylor expanded in x around 0 97.5%

   \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {x}^{2}} \]
9. Step-by-step derivation

   1. *-commutative 97.5%

      \[\leadsto \varepsilon + \color{blue}{{x}^{2} \cdot \varepsilon} \]

   2. distribute-rgt1-in 97.5%

      \[\leadsto \color{blue}{\left({x}^{2} + 1\right) \cdot \varepsilon} \]

   3. unpow2 97.5%

      \[\leadsto \left(\color{blue}{x \cdot x} + 1\right) \cdot \varepsilon \]

   4. fma-define 97.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \cdot \varepsilon \]

10. Simplified 97.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right) \cdot \varepsilon} \]

11. Final simplification 97.5%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x, 1\right) \]
12. Add Preprocessing

Alternative 11: 98.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 eps)

C

double code(double x, double eps) {
	return eps;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps
end function

Java

public static double code(double x, double eps) {
	return eps;
}

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

function tmp = code(x, eps)
	tmp = eps;
end

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 61.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. sub-neg 98.5%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. mul-1-neg 98.5%

      \[\leadsto \varepsilon \cdot \left(1 + \left(-\color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

   3. remove-double-neg 98.5%

      \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

5. Simplified 98.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Taylor expanded in x around 0 96.8%

   \[\leadsto \color{blue}{\varepsilon} \]
7. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (sin eps) (* (cos x) (cos (+ x eps)))))

C

double code(double x, double eps) {
	return sin(eps) / (cos(x) * cos((x + eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin(eps) / (cos(x) * cos((x + eps)))
end function

Java

public static double code(double x, double eps) {
	return Math.sin(eps) / (Math.cos(x) * Math.cos((x + eps)));
}

Python

def code(x, eps):
	return math.sin(eps) / (math.cos(x) * math.cos((x + eps)))

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(cos(x) * cos(Float64(x + eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin(eps) / (cos(x) * cos((x + eps)));
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024139 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (/ (sin eps) (* (cos x) (cos (+ x eps)))))

  (- (tan (+ x eps)) (tan x)))